Statistical decision problem
In mathematical statistics, a statistical decision problem is a triple made up of a statistical model , a decision space and a loss function . Statistical decision problems form the framework for finding optimal decision functions and are therefore a general superstructure for determining good range estimates , point estimates and the design of statistical tests .
definition
A triple is called a statistical decision problem if
- is a statistical model , i.e. for a basic set , a σ-algebra on this basic set and a family of probability measures .
- is the basic set of a decision space , i.e. a measurement space whose σ-algebra contains all point sets.
- is a loss function .
example
Consider the 100-fold product model as a statistical model, which models the repeated coin toss
- ,
The decision space is the measurement space that contains the parameter of the Bernoulli distribution to be estimated and, for example, the Gaussian loss as a loss function
- ,
so is a statistical decision problem.
Statistical decision problems specially tailored to the estimation are also called estimation problems .
literature
- Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .